Prefixes

1. Definitions

A prefix placed before a unit represents a power-of-ten multiplier. For example, 1 km = 10³ m.

2. Key Ideas

Prefix symbols are case-sensitive: m means milli, while M means mega.

3. Detailed Explanations

1. Replace the prefix with its power of ten.
2. Carry out the multiplication.
3. Write the requested unit and sensible significant figures.

Examples:

25 cm = 25 × 10⁻² m = 0.25 m

250 ms = 250 × 10⁻³ s = 0.250 s

Squared and cubed units

Apply the power to the conversion factor as well as the unit:

1 cm² = (10⁻² m)² = 10⁻⁴ m²

1 cm³ = (10⁻² m)³ = 10⁻⁶ m³

Standard form

Standard form is a × 10ⁿ, where 1 ≤ alt 10 and n is an integer. Move the decimal point until the first factor lies in this range, then count the places moved.

Comparing orders of magnitude

An order of magnitude gives the approximate power-of-ten scale of a quantity. It is useful for comparing sizes even when exact values vary. If a question asks you to compare exponent scales, write each value in standard form and compare the exponents. If it instead defines “nearest power of ten”, follow that stated definition.

These are scale comparisons, not exact measurements. A change of one order of magnitude means a factor of about ten.

4. Common Mistakes

• Moving the decimal point from memory without writing the prefix as a power of ten.
• Applying a length conversion only once to an area or volume unit instead of squaring or cubing the conversion factor.
• Keeping a prefixed input and an unprefixed input in the same substitution.

5. Exam Tips

• Replace every prefix by its power of ten before substituting into an equation.
• Put squared or cubed conversion factors in brackets before applying the exponent.
• Convert the final answer to the requested unit and check that the direction of the size change is sensible.

6. Worked Examples

Example 1: Small lengthCore

Convert 13 mum to metres.

Show answer

13 mum = 13 × 10⁻⁶ m = 1.3 × 10⁻⁵ m

Example 2: Volume conversionCore

Convert 40.0 cm³ to cubic metres.

Show answer

40.0 cm³ = 40.0 × (10⁻² m)³ = 4.00 × 10⁻⁵ m³

Example 3: Density conversionCore

Convert 7.8 g cm⁻³ to kg m⁻³.

Show answer

7.8 g cm⁻³ = 7.8 × 10⁻³ kg/10⁻⁶ m³ = 7.8 × 10³ kg m⁻³

Further mistakes to diagnose

• Confusing m (milli) with M (mega).
• Reversing the power when changing unit.
• Squaring or cubing the unit but not its conversion factor.
• Writing standard form with a first factor outside 1 ≤ alt 10.
• Applying a “nearest power of ten” rule when the question asks for the exponent scale, or comparing exponents without first writing both values in standard form.

7. Mind Stretchers

Mind stretcher 1: Convert an areaExtension

A square has side 3.2 cm. Calculate its area in m².

Show answer

Convert the length before squaring:

3.2 cm = 3.2 × 10⁻² m

A = (3.2 × 10⁻²)² = 1.024 × 10⁻³ m²

To two significant figures, A = 1.0 × 10⁻³ m².

Mind stretcher 2: Compare exponent scalesExtension

The radius of Earth is about 6.4 × 10⁶ m and an atom’s diameter is about 1.0 × 10⁻¹⁰ m. Using the exponents in standard form, how many orders of magnitude separate their scales?

Show answer

The exponents are 6 and -10:

6-(-10) = 16

Their exponent scales are separated by 16 orders of magnitude. This does not mean their exact size ratio is precisely 10¹⁶; the coefficients also affect the ratio.

8. Practice, Quiz and Next Step

Close your notes and use Prefixes in the supplied context below. This requires a constructed explanation or working, not recognition of an option.

Fresh context: Convert 3.6 mm² to m² and 2.5 × 10⁶ μs to seconds.

1. Retrieve: define prefix and power-of-ten conversion in your own words, including units, sign or conditions where relevant.
2. Represent: Write each prefix as a power of ten, putting the area conversion in brackets before squaring.
3. Apply: Complete both conversions and use order of magnitude to check the direction of each size change.

Check the response before looking back

• Every numerical value has a justified unit and precision.
• Observed readings, corrections and calculated results are kept distinct.
• The conclusion follows from the instrument or data rather than a memorised label.

If one check fails, name that exact gap, revisit the matching explanation or worked example, and redo the task with different values or a different situation. Then use theO-Level topic checks orpractice browser for an independent re-test.